That is the intention; however, actual agreement is very hard to achieve. In fact, even when you think you have agreement, it is almost impossible to prove that you do. Understanding a language is exactly the same problem as understanding the universe. You may be throughly convinced you understand something and be wrong. And, yes, the problem verges on "paradoxical" but we all have very convincing personal impressions that we understand some things. Every year, millions of babies come into the world understanding not a single word of any language and yet, in a matter of a few short years they all are convinced that they understand a tremendous volume of their experiences. This must be a consequence of a finite number of steps (by the way, infinite means that, no matter how much has been accomplished, you're not finished).
That's what everyone with any sense says; but, as soon as I pressure them as to how that situation should be handled, they suddenly "know" all kinds of things and have no problem to be handled. Your anecdote is a perfect example: it is chock full of things you "know" and is used to avoid facing the fact that it is all assumed.
But communication requires a language, vague and misunderstood as it may be. In this respect, mathematics is the most agreed upon collection of definitions and procedures known to man. If we assume people understand what we are saying when we put down a mathematical relationship we can certainly be more confident than we can with any other set of symbols.
Absolutely everyone, to the last soul, presumes I am presenting a theory. I am not! I am suggesting a procedure for handling the fact that our starting point must be the very absence of understanding of anything. That is why I define the three sets A, B and C
You have not thought that out carefully. One does not actually prove any particular agreement is wrong. What every such proof actually proves is that there is an inconsistency in that proposed agreement . An inconsistency is a case where two different results can be obtained from the same axiomatic starting position. The group is thus forced to readjust their supposed agreement. A proof of error is a proof that the supposed agreement does not actually exit.
Have fun -- Dick